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Dynamic Causal Modelling Revisited

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Dynamic Causal Modelling Revisited

K J Friston et al. Neuroimage.

Abstract

This paper revisits the dynamic causal modelling of fMRI timeseries by replacing the usual (Taylor) approximation to neuronal dynamics with a neural mass model of the canonical microcircuit. This provides a generative or dynamic causal model of laminar specific responses that can generate haemodynamic and electrophysiological measurements. In principle, this allows the fusion of haemodynamic and (event related or induced) electrophysiological responses. Furthermore, it enables Bayesian model comparison of competing hypotheses about physiologically plausible synaptic effects; for example, does attentional modulation act on superficial or deep pyramidal cells - or both? In this technical note, we describe the resulting dynamic causal model and provide an illustrative application to the attention to visual motion dataset used in previous papers. Our focus here is on how to answer long-standing questions in fMRI; for example, do haemodynamic responses reflect extrinsic (afferent) input from distant cortical regions, or do they reflect intrinsic (recurrent) neuronal activity? To what extent do inhibitory interneurons contribute to neurovascular coupling? What is the relationship between haemodynamic responses and the frequency of induced neuronal activity? This paper does not pretend to answer these questions; rather it shows how they can be addressed using neural mass models of fMRI timeseries.

Keywords: Bayesian; Dynamic causal modelling; Effective connectivity; Haemodynamic models; Neural mass models.

Figures

Fig. 1
Fig. 1
Schematic summarising the generative model for each region or node. This model comprises two sets of differential equations modelling neuronal dynamics and haemodynamics respectively. These are coupled via a linear (neurovascular) mapping, such that the neuronal states provide an input to the haemodynamics. Experimental inputs perturb neuronal dynamics that are modelled with a canonical microcircuit. This microcircuit comprises four neuronal populations, comprising spiny stellate cells, superficial pyramidal cells, inhibitory interneurons and deep pyramidal cells. Each population is equipped with two hidden states whose dynamics are described by the second-order ordinary differential equation in the figure. These equations of motion model the depolarisation of each population in response to experimental inputs and afferents from other populations in the same (intrinsic) and other (extrinsic) nodes. The four populations are coupled via intrinsic connections that correspond to known inter-and intralaminar connectivity. Pre-synaptic activity at each subpopulation is then used to drive haemodynamic responses, through local collaterals innervating astrocytes, whose (endfeet) processes release vasodilatory signals. These signals then enter a standard haemodynamic model to generate a BOLD signal. In this graphic, pink connections are inhibitory, blue connections are excitatory and green connections correspond to collateral projections mediating neurovascular responses. Please see Table 1, Table 2 for a list of the variables (and their prior densities). The square brackets are Iverson brackets, returning one when the expression is true and zero otherwise.
Fig. 2
Fig. 2
This figure provide an example of the extrinsic connectivity architecture used in this sort of DCM – and the particular network or graph used in this paper. Here, we have selected three regions that comprise an early visual source (V1), a motion sensitive area (V5 or MST) and an attentional area; the frontal eye fields (FEF). Forward connections arise primarily from superficial pyramidal cells and target spiny stellate cells in the granular layers. In addition, we have modelled a (lower density) connectivity to deep pyramidal cells. Backward connections arise from deep pyramidal cells and target inhibitory interneurons and superficial pyramidal cells. The laminar specificity of these extrinsic connections is specified quantitatively by the prior expectations of the connectivity parameters in the lower equalities. In addition to specifying the extrinsic connectivity architecture, it is necessary to specify where experimental inputs drive or modulate neuronal responses. Here, visual input, visual motion and attention drive responses in the early visual cortex, motion sensitive cortex and frontal eye fields respectively. Crucially, attention exerts a modulatory effect on the self-inhibition of superficial and deep pyramidal cells in the hierarchically intermediate area (V5). Our key question was whether the attentional modulation of superficial, deep or both pyramidal populations is necessary to explain the observed data. Please see the tables for a description of the variables in this figure.
Fig. 3
Fig. 3
This figure shows the results of inference about attentional modulation. The left panel shows the posterior density over the two modulatory effects on superficial and deep pyramidal cells in motion sensitive area V5. The posterior mean is the grey bar, while the 90% posterior confidence intervals are shown in pink. One can see that both (log scaling) effects are substantially greater and smaller than the prior mean of zero (i.e. a scaling of 100%). The intuition that both parameters are necessary to explain the observed responses is confirmed through Bayesian model comparison. The right panel shows the results of Bayesian model reduction of the full model, when eliminating either the modulation of the superficial pyramidal cells, deep pyramidal cells or both. With these data, we can be almost 100% confident that both effects are evident in these data.
Fig. 4
Fig. 4
This figure shows the characterisation of neurovascular coupling in terms of the parameters that couple afferent presynaptic activity to the neurovascular signal in each region. The inset (on the right) shows that inhibitory presynaptic collaterals from the excitatory neuronal populations are the most important. Estimates of the neurovascular coupling parameter are shown on the upper left using the same format as the previous figure. These 12 parameters correspond to intrinsic inhibitory collaterals (shown on dark green) intrinsic excitatory collaterals (shown on green) and extrinsic excitatory collaterals (shown on light green) to each of the four populations. These collaterals provide distinct inputs because the posterior correlations among the associated parameter estimates are not intense (see lower left panel). The Bayesian parameter averages – following Bayesian model reduction over all combinations of the 12 neurovascular coupling parameters – are shown on the upper right. This procedure shrinks redundant parameters to their prior expectation (of zero). The lower right panel shows the posterior probability over all models with and without each of the 12 parameters. These Bayesian parameter averages suggest that, in this instance, intrinsic inhibitory activity is the most important determinant of haemodynamic responses.
Fig. 5
Fig. 5
This schematic illustrates the potential for generating multimodal predictions from the same (neuronal) dynamic causal model. The previous figures focused on the generation of BOLD responses that are mediated by (hidden) neuronal states. However, these states can also be used to generate predictions of the local field potentials or event related responses; here, characterised through a linear mapping with a standard electromagnetic lead field. The first-order kernel mapping from experimental input to predicted electrophysiological responses (at the top of the figure) is what would be seen in response to a very brief stimulus. Under local linearity assumptions, one can use these kernels to predict induced responses that are generated by random fluctuations about the mean neuronal activity. This means that, given the spectral density of neuronal fluctuations, one can generate induced responses. These are illustrated on the right of the figure in terms of the autospectra (with and without observation noise in solid and dashed lines respectively) and the associated autocovariance function (i.e., the Fourier transform of the autospectra). The equations in this figure show the relationships between the first-order kernels, cross spectral density and covariance functions used to generate these sorts of predictions. Please see Table 4 for a full description of the expressions.
Fig. 6
Fig. 6
This figure shows the predicted and observed BOLD responses in each of the regions (upper panel) accompanied by simulated (but unobserved) local field potentials (middle panel). In addition, the time frequency induced responses are shown for the motion sensitive region (V5), over the entire session. The agreement between the predicted (solid lines) and observed (dotted lines) fMRI responses is self-evident. The blue lines correspond to the early visual response (V1) which shows little attentional modulation. Conversely, the motion sensitive area (V5, green lines) shows a profound motion sensitive response that is modulated by attention by about 10%. The frontal eye field responses (red lines) show a marked attentional modulation but little in the way of visual selective responses. The electrophysiological responses show a similar profile; illustrating large offset and onset responses and then maintenance at the fixed point for each level of experimental input. The attentional modulation of the superficial and deep pyramidal cells in the motion sensitive area changes the connectivity and subsequent predictions of induced responses. These are entirely consistent with alpha (at 10 Hz) desynchronization during attention that is accompanied by an increase in gamma activity (at 48 Hz). The genesis of these induced responses is addressed in more detail in the next figure. The light green bars indicate periods of attention to visual motion.
Fig. 7
Fig. 7
This figure shows the effect of modulating the self-inhibition of each of the four subpopulations (in the absence of afferent or experimental input). Each row shows the autospectra from each of the four populations (spiny stellate, superficial pyramidal, inhibitory interneurons and deep pyramidal cells respectively) over a log scaling from −2 to +2. The left panels show the resulting autospectra from 0 to 96 Hz, while the right panels show the same data in image format. These suggest that increasing the self-inhibition of spiny stellate cells rapidly suppresses alpha activity and increases the frequency of gamma activity until a bifurcation at a peak gamma activity of about 80 Hz. This phase transition is seen even earlier as the self-inhibition of superficial pyramidal cells increases, with a peak gamma of about 42 Hz. The effects of increasing self-inhibition of inhibitory interneurons and deep pyramidal cells are to suppress alpha activity and convert it into fast activity. See main text for further discussion.
Fig. 8
Fig. 8
This figure highlights the (degenerate) relationship between fluctuations in spectral power and haemodynamic responses. The three rows of this figure report the responses of the three regions; namely, the early visual region, a motion sensitive region and the frontal eye fields. The left panels show the first principal component or eigenvariate of fluctuations in the power of induced responses (based upon the posterior estimates of attentional modulation in Fig. 3). The right panels plot observed haemodynamic responses against the expression of these frequency modes. The dotted lines connect consecutive time points. In the early visual cortex, there is a profound alpha suppression that is accompanied by an increase in gamma in V5. This desynchronization is limited to gamma activity in the FEF.

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